We study the twisted cohomoligical equation over the geodesic flow on $ SL(2, \mathbb{R} )/\Gamma $. We characterize the obstructions to solving the twisted cohomological equation, construct smooth solution and obtain the tame Sobolev estimates for the solution, i.e, there is finite loss of regularity (with respect to Sobolev norms) between the twisted coboundary and the solution. We also give a tame splittings for non-homogeneous cohomological equations. The result can be viewed as a first step toward the application of KAM method in obtaining differential rigidity for partially hyperbolic actions in products of rank-one groups in future works.

Samuel C. Edwards.
On the rate of equidistribution of expanding horospheres in finite-volume quotients of SL(2, ${\mathbb{C}}$).
Journal of Modern Dynamics,
2017, 11: 155-188.
doi: 10.3934/jmd.2017008

[8]

Anne-Sophie de Suzzoni.
Consequences of the choice of a particular basis of $L^2(S^3)$ for the cubic wave equation on the sphere and the Euclidean space.
Communications on Pure & Applied Analysis,
2014, 13
(3)
: 991-1015.
doi: 10.3934/cpaa.2014.13.991